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This module offers a brief and generally intuitive introduction to maximum likelihood estimation methods. It is intended as a guide for advanced undergraduates.

The maximum likelihood method

Introduction

The maximum likelihood (ML) method is an alternative to ordinary least squares (OLS) and offers a more general approach to the problem of finding estimators of unknown population parameters. In these notes we present an intuitive introduction to the ML technique. We begin our discussion with a description of continuous random variables.

Continuous random variables

Assume that x MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F1@ is a continuous random variable over the interval x . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeyOhIuQaeyizImQaamiEaiabgsMiJkabg6HiLkaac6caaaa@3EDC@ Because of the assumption of continuity we need some special definitions.

Probability density function . Any function f ( x ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaaaa@3965@ that has the following characteristics is a probability density function (pdf): (1) f ( x ) > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg6da+iaaicdaaaa@3B27@ and (2) f ( x ) d x = 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0JaaGymaiaac6caaaa@4400@ The probability that x has a value between a and b is given by Pr ( a x b ) = a b f ( x ) d x . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadggacqGHKjYOcaWG4bGaeyizImQaamOyaaGaayjkaiaawMcaaiabg2da9maapehabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaaiOlaaaa@4ACB@ Here are two examples of the probability density functions (pdf) of continuous random variables.

Uniform distribution

Let f ( x ) = 1 α MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeg7aHbaaaaa@3CD5@ for 0 x α MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadIhacqGHKjYOcqaHXoqyaaa@3CB4@ and 0 elsewhere, where α > 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOpa4JaaGimaiaac6caaaa@3A07@ A graph of the pdf for this distribution is shown in Figure 1.

Probability distribution function of a uniform distribution.

A graph of the uniform distribution.
The probability x falls between a and b is given by the colored in area.

It is easy to see from the graph that f ( x ) = 1 α > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeg7aHbaacqGH+aGpcaaIWaaaaa@3E97@ and Pr ( a x b ) = f ( x ) d x = 0 α 1 α d x = 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaabccaciGGqbGaaiOCamaabmaabaGaamyyaiabgsMiJkaadIhacqGHKjYOcaWGIbaacaGLOaGaayzkaaGaeyypa0Zaa8qCaeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0Zaa8qCaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaGaamizaiaadIhaaSqaaiaaicdaaeaacqaHXoqya0Gaey4kIipakiabg2da9iaaigdacaGGUaaaaa@59F7@ Moreover, as shown in Figure 1, the area under the pdf curve between a and b is equal to the probability that x lies between a and b ; that is, Pr ( a x b ) = a b ( 1 α ) d x = x α | a b = b a α . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadggacqGHKjYOcaWG4bGaeyizImQaamOyaaGaayjkaiaawMcaaiabg2da9maapehabaWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaaacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdGccqGH9aqpdaabcaqaamaalaaabaGaamiEaaqaaiabeg7aHbaaaiaawIa7amaaDaaaleaacaWGHbaabaGaamOyaaaakiabg2da9maalaaabaGaamOyaiabgkHiTiaadggaaeaacqaHXoqyaaGaaiOlaaaa@5809@

The calculation of the mean and variance of this distribution is relatively simple. The population mean is given by μ x = E ( x ) = 0 α x f ( x ) d x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaSbaaSqaaiaadIhaaeqaaOGaeyypa0JaamyramaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maapehabaGaamiEaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaWcbaGaaGimaaqaaiabeg7aHbqdcqGHRiI8aaaa@494E@ or μ x = 0 α x ( 1 α ) d x = x 2 2 α | 0 α = α 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaSbaaSqaaiaadIhaaeqaaOGaeyypa0Zaa8qCaeaacaWG4bWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaaacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiaaicdaaeaacqaHXoqya0Gaey4kIipakiabg2da9maaeiaabaWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiabeg7aHbaaaiaawIa7amaaDaaaleaacaaIWaaabaGaeqySdegaaOGaeyypa0ZaaSaaaeaacqaHXoqyaeaacaaIYaaaaaaa@527C@

The population variance Quite often, as in the exercises at the end of this module, it is easier to calculate the variance of a distribution using the alternative formula for the variance: σ x 2 = V ( x ) = E ( x μ ) 2 = E ( x 2 ) μ 2 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDaaaleaacaWG4baabaGaaGOmaaaakiabg2da9iaadAfadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGfbWaaeWaaeaacaWG4bGaeyOeI0IaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamyramaabmaabaGaamiEamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgkHiTiabeY7aTnaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@4F7E@ where E ( x 2 ) = x 2 f ( x ) d x . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacqGH9aqpdaWdbaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqabeqaniabgUIiYdGccaGGUaaaaa@4530@ is given by V ( x ) = E [ ( x μ x ) 2 . ] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadweadaWadaqaamaabmaabaGaamiEaiabgkHiTiabeY7aTnaaBaaaleaacaWG4baabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5waiaaw2faaaaa@4465@ Thus, V ( x ) = 0 α ( x α 2 ) 2 ( 1 α ) d x = 0 α ( x 2 α x + α 2 4 ) ( 1 α ) d x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@62B5@ or V ( x ) = x 3 3 α x 2 2 + α 4 x | 0 α = α 2 3 α 2 2 + α 2 4 = α 2 12 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5D95@

Because of the simple mathematical form of the uniform pdf, the calculations in Example 1 are relatively straight forward. While the calculations for random variables with a pdf that has a more complicated form are generally more difficult (if algebraically possible), the basic methodology remains the same. Example 2 considers the case of a more complicated pdf.

The normal distribution.

A random variable with a mean of μ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AA@ and a variance of σ 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaa@38A0@ that has a normal distribution —that is, x ~ N ( μ , σ 2 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaac6hacaWGobWaaeWaaeaacqaH8oqBcaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiifGaaa@4023@ has the pdf f ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeo8aZnaakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaWcaaqaamaabmaabaGaamiEaiabgkHiTiabeY7aTbGaayjkaiaawMcaamaaCaaameqabaGaaGOmaaaaaSqaaiaaikdacqaHdpWCdaahaaadbeqaaiaaikdaaaaaaaaakiaac6caaaa@4BED@ A typical graph of this pdf is given in Figure 2. The area under the curve between values of x of a and b is equal to the probability that x falls between a and b .

Probability distribution function of a normal distribution.

A graph of the Normal distribution.
The probability x falls between a and b is given by the shaded area.

Joint distributions of samples and the ml method.

Most of the statistical work that economists use involves the use of a sample of observations. It is usual to assume that the members of the sample are drawn independently of each other. The implication of this assumption is that the pdf of the joint distribution is equal to the product of the pfd of each observation ; i.e.,

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Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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